Our estimates give
\(&\qquad \qquad \quad \int _{T}^{2T} |\zeta (\tfrac{1}{2} + i t)|^{2k - 2} |\zeta ^{\prime }(\tfrac{1}{2} + i t)|^2 dt \ll \\T(\log T&)^{k^2 + 2} + \sum _{2\le v \le \ell } T (\log T_{v-1} )^{k^2 + 2} \Bigg ( \left(\frac{\log T}{\log T_{v-1} }\right)^3 \frac{2^{\lceil 50 P_v \rceil } \lceil 50 P_v \rceil ! P_v^{\lceil 50 P_v \rceil }\exp (P_v)}{(50 P_v)^{2\lceil 50 P_v \rceil }} \\&+ \left(\frac{\log T}{\log T_{v-1} }\right)^6 \frac{18^{\lceil 50 P_v \rceil } \lceil 50 P_v \rceil ! P_v^{\lceil 50 P_v \rceil }\exp (P_v)}{(50 P_v)^{2\lceil 50 P_v \rceil }} \Bigg ) \ll T (\log T)^{k^2 + 2},\) 

where the final bound follows by the same reasoning as [1]}. The conclusion for the \(Z\)  function is the same.
